Real time missile guidance system

Abstract

A weak Hamiltonian finite element method is used for iterative computation of missile guidance acceleration commands for maximizing a missile's terminal velocity while satisfying control authority limits and terminal attitude constraints. The guidance acceleration commands include commands for controlling the angle of attack (α) and the bank angle (φ) of the missile. The angle of attack (α) and bank angle (φ) are related to a set of virtual control variables selected to avoid convergence problems when the angle of attack is approximately zero. The preferred control variables are β 2 and β 3 such that β 2 =cosφtanα and β 3 =sinφtanα. Iterative convergence is facilitated when control inequality constraint parameters are reached by adjusting iterative solutions between iterations toward satisfaction of the constraints. An approximation to an optimal trajectory is calculated at each guidance cycle during missile flight using data which are revised during each guidance cycle. The revised data include current position data for the target and the current position for the missile. The revised data are taken from the most reliable source currently available, such as on-board target-seeking radar when the target-seeking radar is locked onto the target, uplink data from ground or airborne tracking radar when an uplink is operational, or inertial guidance data. Extracted from the optimal trajectory is an optimal acceleration command for optimally controlling the angle of attack and bank angle of the missile.

What is claimed is: 
     
       1. A method of real-time guidance and control of a missile by computing control commands in a data processor during flight of said missile to guide said missile along a trajectory that will optimize a predetermined performance function, said control commands adjusting an angle of attack (α) between a longitudinal axis of said missile and a relative wind vector, and a bank angle (φ) between a transverse axis of said missile and said relative wind vector, said method periodically performing computational cycles in a data processor to compute a set of values of said control commands during each cycle based on a current missile velocity vector, and a target intercept position relative to a current position of said missile, wherein the computations during each cycle include the steps of: a) computing new values for a set of control variables based on said performance function and said current missile velocity vector, and said target intercept position relative to said current missile position; and   b) computing said control commands from said new values of said set of control variables;   wherein said control variables include at least two control variables that together specify said angle of attack (α) and said bank angle (φ), but which have definite and limited values when specifying an angle of attach (60 ) of approximately zero.   
     
     
       2. The method as claimed in claim 1, wherein said at least two control variables include a first control variable (β 2 ) that is a cosine function of the specified bank angle (φ), and a second control variable (β 3 ) that is a sine function of the specified bank angle (φ), so that the ratio (β 3  /β 2 ) of said second control variable to said first control variable is proportional to the tangent of the specified bank angle (φ). 
     
     
       3. The method as claimed in claim 2, wherein said first control variable (β 2 ) and said second control variable (β 3 ) are related to the specified angle of attack (α) and the specified bank angle (φ) as β 2  =cosφ tanα and β 3  =sinα tanα. 
     
     
       4. The method as claimed in claim 1, wherein said step (a) of computing new values for said set of control variables is performed iteratively and wherein a new iterative solution (X.sup.[k+1]) is obtained as a predetermined solution function (S(X.sup.[k])) of a previous iterative solution (X.sup.[k]) beginning with a predetermined trial solution (X.sup.[0]). 
     
     
       5. The method as claimed in claim 4, wherein said previous iterative solution (X.sup.[k]) computed from said predetermined solution function is checked for violation of an inequality constraint, and said previous iterative solution is adjusted toward satisfying said inequality constraint before being used in said solution function to compute said new iterative solution (X.sup.[k+1]). 
     
     
       6. The method as claimed in claim 5, wherein said solution function is predetermined to solve a system of algebraic equations including a summation over finite elements of time, and said inequality constraint specifies that a duration (Δt) of each of said finite elements of time is at least a predetermined positive value. 
     
     
       7. The method as claimed in claim 4, wherein said solution function is predetermined to solve a system of algebraic equations resulting from taking a first variation of an integral of said performance function along a projected trajectory of said missile to said target intercept position, setting said first variation to zero, and approximating an integral in said first variation as a summation over finite elements, wherein said integral of said performance function includes constraints adjoined to said integral of said performance function by Lagrangian multipliers. 
     
     
       8. The method as claimed in claim 7, wherein said performance function is selected to maximize a terminal velocity of said missile at said target intercept position, and said constraints include an angular orientation of said target intercept position. 
     
     
       9. The method as claimed in claim 4, wherein said solution function is a function of position of a moving target, and said solution function is updated between said computational cycles in response to variation in said position of said moving target. 
     
     
       10. A method of real-time guidance and control of a missile by computing control commands in a data processor during flight of said missile, said method including the steps of periodically performing computational cycles in a data processor to compute a set of values of said control commands during each cycle based on equations of motion for said missile and position of a target relative to position of said missile, wherein the computations during each cycle include an iterative computation of new values for control variables, wherein a new iterative solution (X.sup.[k+1]) is obtained as a predetermined solution function (S(X.sup.[k])) of a previous iterative solution (X.sup.[k]) beginning with a predetermined trial solution (X.sup.[0]), and wherein said previous iterative solution (X.sup.[k]) computed from said predetermined solution function is checked for violation of an inequality constraint, and said previous iterative solution is adjusted toward satisfying said inequality constraint before being used in said solution function to compute said new iterative solution (X.sup.[k+1]), and wherein said control commands are computed from said control variables. 
     
     
       11. The method as claimed in claim 10, wherein said previous iterative solution is adjusted toward satisfying said inequality constraint before being used in said solution function when said previous iterative solution is found to violate said inequality constraint by selectively moving a control variable either on or off of a control boundary of said inequality constraint so that said inequality constraint is satisfied. 
     
     
       12. The method as claimed in claim 10, wherein said solution function is predetermined to solve a system of algebraic equations including a summation over finite elements of time, and said inequality constraint specifies that a duration (Δt) of each of said finite elements of time is at least a predetermined positive value. 
     
     
       13. The method as claimed in claim 10, wherein said control commands adjust an angle of attack (α) between a longitudinal axis of said missile and a relative wind vector, and a bank angle (φ) between a transverse axis of said missile and said relative wind vector, and wherein said control variables specify said angle of attack (α) and said bank angle (φ) so that said control variables assume definite and limited values when specifying a value of approximately zero for said angle of attack (α). 
     
     
       14. The method as claimed in claim 13, wherein said control variables include a first control variable (β 2 ) that is a cosine function of the specified bank angle (φ), and a second control variable (β 3 ) that is a sine function of the specified bank angle (φ), so that the ratio (β 3  /β 2 ) of said second control variable to said first control variable is proportional to the tangent of the specified bank angle (φ). 
     
     
       15. The method as claimed in claim 14, wherein said first control variable (β 2 ) and said second control variable (β 3 ) are related to the specified angle of attack (α) and the specified bank angle (φ) as β 2  =cosφ tanα and β 3  =sinφ tanα. 
     
     
       16. The method as claimed in claim 10, wherein said control commands guide said missile along a trajectory that will optimize a predetermined performance function, and said solution function is predetermined to solve a system of algebraic equations resulting from taking a first variation of an integral of said performance function along a projected trajectory of said missile, setting said first variation to zero, and approximating an integral in said first variation as a summation over finite elements, wherein said integral of said performance function includes inequality constraints adjoined to said integral of said performance function by Lagrangian multipliers. 
     
     
       17. The method as claimed in claim 16, wherein said integral of said performance function includes a terminal target constraint adjoined to said integral of said performance function by a discrete Lagrangian multiplier, said performance function is selected to maximize a terminal velocity of said missile upon interception with an intercept position of said target, and said terminal target constraint includes an angular orientation of said missile upon arrival at said intercept position of said target. 
     
     
       18. A method of real-time guidance and control of a missile by computing control commands in a data processor during flight of said missile to guide said missile along a trajectory that will optimize a predetermined performance function, said method including the steps of periodically performing computational cycles in a data processor to compute a set of values of said control commands during each cycle based on equations of motion for said missile and position of a target relative to position of said missile, wherein: the computations during each cycle include an iterative computation of new values for control variables, wherein a new iterative solution (X.sup.[k+1]) is obtained as a predetermined solution function (S(X.sup.[k])) of a previous iterative solution (X.sup.[k]) beginning with a predetermined trial solution (X.sup.[0]),   said control commands are computed from said control variables;   said solution function is predetermined to solve a system of algebraic equations resulting from taking a first variation of an integral of said performance function along a projected trajectory of said missile, setting said first variation to zero, and approximating an integral in said first variation as a summation over finite elements, wherein said integral of said performance function includes an integrand component (L) and an inequality constraint function (G) adjoined to said integral of said performance function by a Lagrangian multiplier (μ), and   said control commands adjust an angle of attack (α) between a longitudinal axis of said missile and a relative wind vector, and a bank angle (φ) between a transverse axis of said missile and said relative wind vector, and wherein said control variables are selected to specify said angle of attack (α) and said bank angle (φ) so that said control variables assume definite and limited values when specifying a value of approximately zero for said angle of attack, and wherein said equations of motion are defined by state functions (f), and said control variables are selected to avoid negative eigenvalues in the matrix ##EQU55## where i and k are indexed over said control variables, j is indexed over said control variables, and k is indexed over said state functions and said inequality constraint function such that:   H=λ.sup.T f+L+μ.sup.2.spsp.T G     wherein λ is a matrix of unknown Lagrangian multiplier functions for adjoining the state functions (f), and T denotes a matrix transpose operation.     
     
     
       19. The method as claimed in claim 18, wherein said control variables include a first control variable (β 2 ) that is a cosine function of the specified bank angle (φ), and a second control variable (β 3 ) that is a sine function of the specified bank angle (φ), so that the ratio (β 3  /β 2 ) of said second control variable to said first control variable is proportional to the tangent of the specified bank angle (φ). 
     
     
       20. The method as claimed in claim 19, wherein said first control variable (β 2 ) and said second control variable (β 3 ) are related to the specified angle of attack (α) and the specified bank angle (φ) as β 2  =cosφ tanα and β 3  =sinφ tanα. 
     
     
       21. The method as claimed in claim 18, wherein said previous iterative solution (X.sup.[k]) computed from said predetermined solution function is checked for violation of said inequality constraint function, and said previous iterative solution is adjusted toward satisfying said inequality constraint function before being used in said solution function to compute said new iterative solution (X.sup.[k+1]). 
     
     
       22. The method as claimed in claim 21, wherein said previous iterative solution is adjusted toward satisfying said inequality constraint function before being used in said solution function when said previous iterative solution is found to violate said inequality constraint function by selectively moving one of said control variables either on or off of a control boundary of said inequality constraint function so that said inequality constraint function is satisfied. 
     
     
       23. The method as claimed in claim 22, wherein a slack variable. (K) is used in adjoining said inequality constraint function (G) to said integral of said performance function, a new value for said slack variable (K) is computed before being used in said solution function when said control variable is moved off of said control boundary, and wherein a new value for said Lagrangian multiplier (μ) is computed before being used in said solution function when said control variable is moved off of said control boundary. 
     
     
       24. The method as claimed in claim 21, wherein said system of algebraic equations includes a summation over finite elements of time, said previous iterative solution includes a duration (Δt) of each of said finite elements of time, and when said duration (Δt) of said previous iterative solution is negative, said duration is set to a positive value before being used in said solution function to compute said new iterative solution. 
     
     
       25. The method as claimed in claim 18, wherein said integral of said performance function includes a terminal constraint adjoined to said integral of said performance function by a discrete Lagrangian multiplier, said performance function having been selected to maximize a terminal velocity of said missile upon interception with said target, and said terminal constraint includes an angular orientation of said missile upon said interception with said target. 
     
     
       26. The method as claimed in claim 18, wherein said solution function is a function of position of said target, and said solution function is updated between said computational cycles in response to variation in said position of said target. 
     
     
       27. A missile comprising an inertial measurement unit and a control system for maneuvering said missile in response to control commands to guide said missile along a trajectory that will optimize predetermined performance function, and a data processing unit connected to said control system to transmit said control commands to said control system, said data processor being programmed for periodically computing said control commands during computational cycles by computing a set of values of said control commands during each cycle based on equations of motion for said missile, position of a target relative to position of said missile, and attitude of said missile, said data processor being programmed to perform during each cycle an iterative computation of new values for control variables, wherein a new iterative solution (X.sup.[k+1]) is obtained as a predetermined solution function (S(X.sup.[k])) of a previous iterative solution (X.sup.[k]) beginning with a predetermined trial solution (X.sup.[0]), and said control commands are computed from said control variables, wherein said solution function is predetermined to solve a system of algebraic equations resulting from taking a first variation of an integral of said performance function along a projected trajectory of said missile, setting said first variation to zero, and approximating an integral in said first variation as a summation over finite elements, wherein said integral of said performance function includes an integrand component (L) and an inequality constraint function (G) adjoined to said integral of said performance function by a Lagrangian multiplier (μ), and at least one discrete Lagrangian multiplier adjoined to said integral of said performance function for specifying a terminal constraint on said attitude of said missile upon interception with said target, and wherein said control commands adjust an angle of attack (α) between a longitudinal axis of said missile and a relative wind vector, and a bank angle (φ) between a transverse axis of said missile and said relative wind vector, and wherein said control variables are selected to specify said angle of attack (α) and said bank angle (φ) so that said control variables assume definite and limited values when specifying a value of approximately zero for the angle of attack (α), and wherein said processor is programmed for checking said previous iterative solution X.sup.[k] computed from said predetermined solution function for violation of said inequality constraint function, and adjusting said previous iterative solution toward satisfying said inequality constraint function before being used in said solution function to compute said new iterative solution X.sup.[k+1]. 
     
     
       28. The method as claimed in claim 27, wherein said control variables include a first control variable (β 2 ) that is a cosine function of the specified bank angle (φ), and a second control variable (β 3 ) that is a sine function of the specified bank angle (φ), so that the ratio (β 3  /β 2 ) of said second control variable to said first control variable is proportional to the tangent of the specified bank angle (φ). 
     
     
       29. The missile as claimed in claim 28, wherein said first control variable (β 2 ) and said second control variable (β 3 ) are related to the specified angle of attack (α) and the specified bank angle (φ) as β 2  =cosφ tanα and β 3  =sinφ tanα. 
     
     
       30. The missile as claimed in claim 27, wherein said data processor is programmed to adjust said previous iterative solution toward satisfying said inequality constraint function before being used in said solution function when said previous iterative solution is found to violate said inequality constraint function by selectively moving a control variable either on or off of a control boundary of said inequality constraint function so that said inequality constraint function is satisfied. 
     
     
       31. The method as claimed in claim 30, wherein a slack variable (K) is used in adjoining said inequality constraint function (G) to said integral of said performance function, and said data processor is programmed to compute a new value for said slack variable (K) before being used in said solution function when said control variable is moved off of said control boundary, and to compute a new value for said Lagrangian multiplier (μ) before being used in said solution function when said control variable is moved on said control boundary. 
     
     
       32. The missile as claimed in claim 27, further comprising an uplink receiver connected to said data processor for providing uplinked tracking data indicating the position of said target relative to the position of said missile during flight of said missile for use in said computational cycles. 
     
     
       33. The missile as claimed in claim 27, wherein said performance function has been selected to maximize a terminal velocity of said missile upon interception with said position of said target.